Bruce L. Golden

R.H. Smith School of Business

University of Maryland

Presented at AIRO 2012 Conference, September 2012

Vietri sul Mare, Italy

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Outline of Talk Some personal remarks

Vehicle Routing

The Hierarchical Traveling Salesman Problem (HTSP)

Healthcare Analytics

The Effects of Bed Utilization on Discharge and Readmission Rates

Conclusions

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Grand Hotel Convento di Amalfi

Introduction to the HTSP Consider the distribution of relief aid

E.g., food, bottled water, blankets, or medical packs

The goal is to satisfy demand for relief supplies at many locations

Try to minimize cost

Take the urgency of each location into account

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A Simple Model for Humanitarian Relief Routing Suppose we have a single vehicle which has enough

capacity to satisfy the needs at all demand locations from a single depot

Each node (location) has a known demand (for a single product called an aid package) and a known priority

• Priority indicates urgency

• Typically, nodes with higher priorities need to be

visited before lower priority nodes

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Node Priorities Priority 1 nodes are in most urgent need of service

To begin, we assume

• Priority 1 nodes must be served before priority 2

nodes

• Priority 2 nodes must be served before priority

3 nodes, and so on

• Visits to nodes must strictly obey the node

priorities

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The Hierarchical Traveling Salesman Problem We call this model the Hierarchical Traveling

Salesman Problem (HTSP)

Despite the model’s simplicity, it allows us to explore the fundamental tradeoff between efficiency (distance) and priority (or urgency) in humanitarian relief and related routing problems

A key result emerges from comparing the HTSP and TSP in terms of worst-case behavior

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Four Scenarios for Node Priorities

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Literature Review Psaraftis (1980): precedence constrained

TSP

Fiala Tomlin, Pulleyblank (1992): precedence constrained helicopter routing

Guttman-Beck et al. (2000): clustered traveling salesman problem

Campbell et al. (2008): relief routing

Balcik et al. (2008): last mile distribution

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A Relaxed Version of the HTSP Definition: The d-relaxed priority rule adds

operational flexibility by allowing the vehicle to visit nodes of priority π + 1, … , π + d (if these priorities exist in the given instance) but not priority π + d + ℓ for ℓ ≥ 1 before visiting all nodes of priority π (for π = 1, 2,…,P)

When d=0, we have the strict HTSP

When d=P-1, we have the TSP (i.e., we can ignore node priorities)

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Efficiency vs. Priority

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Main Results Let P be the number of priority classes

Assume the triangle inequality holds

Let Z∗d,P and Z∗

TSP be the optimal tour length (distance) for the HTSP with the d-relaxed priority rule and for the TSP (without priorities), respectively

We obtain the following results

(a)

(b) Z d,P∗ ≤

P

d+1 ZTSP

∗

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Z∗0,P ≤ PZ∗

TSP

Sketch of Proof (a)

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Tour τ* Length = Z*TSP

Sketch of Proof (a)

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Sketch of Proof of (b)

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Tour τ* Length = Z*TSP

Tour τ* Length = Z*TSP

Sketch of Proof of (b)

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The General Result and Two Special Cases

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Worst-case Example

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Several Observations Observation 1. The worst-case example shows that the

bounds in (a) and (b) are tight and cannot be improved

Observation 2. We can “solve” a TSP over the entire set of nodes using our favorite TSP heuristic and obtain a feasible tour for the HTSP by following the part (b) proof

Observation 3. Suppose we select Christofides’ heuristic

and let 𝑍𝑑,𝑃ℎ be the length of the resulting feasible solution

to the HTSP, then we have 𝑍𝑑,𝑃ℎ ≤

3

2∙

𝑃

𝑑+1𝑍𝑇𝑆𝑃

∗

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Observations and Extensions Observation 4. The HTSP (with d=0) can be modeled

and solved as an ATSP

Observation 5. Other applications of the HTSP include routing of service technicians and routing of unmanned aerial vehicles

We can obtain similar worst-case results (with tight bounds) for the HTSP on the line and the Hierarchical Chinese Postman Problem (HCPP)

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Extensions and Future Work The HTSP and several generalizations have been

formulated as mixed integer programs

HTSP instances with 30 or so nodes were solved to optimality using CPLEX

Future work

The Hierarchical Vehicle Routing Problem (HVRP)

A multi-day planning horizon

Uncertainty with respect to node priorities

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Emergence of Healthcare Analytics within INFORMS

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0

50

100

150

200

250

300

Above numbers courtesy of Brian Denton

Number of Healthcare Talks at INFORMS Annual Meetings

Strength in Numbers There is more healthcare data available than ever

before

Careful analysis of healthcare data can lead to smarter decisions, better quality healthcare, and cost savings

A larger number of healthcare decision makers have MBAs than ever before

They understand that we can help

A larger number of us in OR/OM are working on healthcare applications than ever before

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The Effects of Bed Utilization on Discharge and Readmission Rates Many hospital resources are required for surgery

Operating rooms

Nurses & Physicians

Anesthesia team

Post-operative beds for recovery

If downstream beds are unavailable, surgery might be postponed or cancelled

Surgeons decide when patients are discharged Surgeons are paid to do surgery

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Research Question 1 Does the utilization of downstream beds

affect the discharge decisions of surgeons?

Hypothesis: There is an increased discharge rate on days when post-operative utilization is high

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Data Data collected on every surgery performed at a large

US hospital from Jan 1, 2007 to May 31, 2007

7808 patients, of which 6470 were admitted to the hospital and stayed for at least one night

These patients stayed a total of 35,478 days

Data provided on age, race, gender, surgical line, date of surgery, discharge date, and surgery type (scheduled vs. emergency)

Utilization of post-operative beds varies widely

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Discharge Rates

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Discharge rates have positive correlation with utilization

Utilization Measures We compute two measures of utilization

Discrete measure – a variable that is 1 when utilization exceeds a given threshold (e.g., 93%), and 0 otherwise

Continuous measure – a variable that counts the number of beds in use on each day

Compare marginal effect of each bed in use vs. a discrete change in discharge probability when utilization exceeds a threshold

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Discrete Time Survival Analysis Can’t use logistic regression because observations are correlated -- a

patient discharged on the fifth day cannot be discharged on the first four days

Singer and Willet (1993) show how to handle discrete time survival data

For each day, we record whether or not each patient is discharged, and use this as the outcome variable

The outcome variable is regressed on our utilization measures and our control variables

We control for the patient’s age, race, gender, severity, and surgery type

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Models and Results Model 1: logit(DISCHARGE) = AGE + ELECTIVE + FULL + CARDIAC SURGERY

+ CARDIOLOGY +…+ DONOR SERVICE + D1

+ D2 +…+ D59 + ε

Model 2: logit(DISCHARGE) = AGE + ELECTIVE + BEDS + CARDIAC SURGERY

+ CARDIOLOGY +…+ DONOR SERVICE + D1

+ D2 +…+ D59 + ε

When the utilization threshold is exceeded, the odds of discharge for any given patient increase. The estimate for Full is positive and significant for threshold above 91.5%.

Each additional bed in use increases the odds that a patient will be discharged. The estimate for Beds is positive and significant.

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Observations Discharge rates increase as utilization increases,

regardless of how utilization is measured

Either some patients are held too long and discharged when space is needed, or some patients are discharged too early when utilization is high

Our results cannot distinguish between these two explanations

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Research Question 2 Are patients who are discharged when utilization

is high more likely to be readmitted?

Hypothesis: An increase in the discharge rate will lead to some patients with shortened lengths of stay. This will cause an increase in the readmission rate for those patients.

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Analysis Using the same data set, we apply logistic regression to

study the effect that utilization has on the probability of readmission for a specific patient

We use readmission within 72 hours as our dependent variable

Hypothesized logistic regression model

logit(READMISSION72) = AGE + BLACK + ASIAN + HISPANIC + FULL (or BEDS)+ ELECTIVE + TRANSPLANT + TRAUMA + … + NEURO + MALE + ε

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Results Model with Full: Controlling for race, age, gender, and

the type of surgery, being discharged from a full post-operative unit increases the odds of readmission by a factor of 2.341

Model with Beds: Controlling for race, age, gender, and the type of surgery, each bed in use at the time of discharge increases the odds of readmission by a factor of 1.008

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Utilization-Readmission Relationship

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The discharge rate

and readmission

rate both increase

as utilization

increases

Survival Analysis

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Over the course of a

month, patients

discharged from a full

hospital are much more

likely to be readmitted

Discussion The discharge rate rises when utilization is high

This corresponds to an increase in the readmission rate

We conclude that some patients are discharged too soon when utilization is high

Surgeons have an incentive to clear space for their surgeries

Mitigation strategy: Use a checklist before discharging a patient—force the surgeon to think about whether the discharge is for the right reason

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Conclusions Research opportunities in vehicle routing, disaster

relief, and healthcare analytics are plentiful

The HTSP work presented here will appear in Optimization Letters

The healthcare analytics work presented here has appeared in Health Care Management Science (2011, 2012)

Thank you!

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